How to calculate the definite integral of the complex function $\frac{1}{(a+ib + (a-ib)x^2)}$ for $x\in(-\infty,\infty)$

Question

How do I calculate the following integral:

$\int_{-\infty}^{\infty} \frac{1}{a+bi+(a-bi)x^2}dx$,

where $a,b\in\mathbb{R}$.

Answer 1

Let $$a+ib=r e^{ic}, r=\sqrt{a^2+b^2},~\frac{a+ib}{a-ib}=e^{2ic},$$ then $$I=\frac{1}{a-ib} \int_{-\infty}^{\infty} \frac{dx}{x^2+e^{2ic}} =\frac{e^{ic}}{r} e^{-ic} \tan^{-1}x e^{-ic}|_{-\infty}^{\infty}=\frac{\pi}{\sqrt{a^2+b^2}}$$